Nonlinear harmonics of a seeded free-electron laser as a coherent and ultrafast probe to investigate matter at the water window and beyond

The advent of free-electron lasers (FELs) in the soft- and hard-x-ray spectral regions has introduced the possibility to probe electronic, magnetic, and structural dynamics, in both diluted and condensed matter samples, with femtosecond time resolution. In particular, FELs have strongly enhanced the capabilities of several analytical techniques, which have taken advantage of the high degree of transverse coherence provided. Free-electron lasers based on the harmonic up-conversion of an external coherent source (seed) are characterized also by a high degree of longitudinal coherence, since electrons inherit the coherence properties of the seed. For the state of the art, the shortest wavelength delivered to user experiments by an externally seeded FEL light source is about 4 nm. In this paper we demonstrate that pulses with a high longitudinal degree of coherence (first and second order) covering the water window and with photon energy extending up to 790 eV can be generated by exploiting the so-called nonlinear harmonic regime, which allows generation of radiation at harmonics of the resonant FEL wavelength. In order to show the suitability of the nonlinear harmonics generated by a seeded FEL for research in the water window and beyond, we report the results of two proof-of-principle experiments: one measuring the oxygen $K$-edge absorption in water ($\sim 530$ eV) and the other analyzing the spin dynamics of Fe and Co through magnetic small-angle x-ray scattering at their $L$ edges (707 and 780 eV, respectively).